if three no.s are added their sum is 2 .if 2 times the second no.s is subtracted from the sum of first and third no.is added to the sum of second and third no.we get 4 find the no, s using matrices
Answers
Answer:
ANSWER
Let three numbers be x,y,z.
From given conditions, we have
x+y+z=2 .............(1)
x+z−2y=8
x−2y+z=8 ...........(2)
And
3x+y+z=4 .........(3)
Given all equation can be written in matrix form as,
⎣
⎢
⎢
⎡
1
1
3
1
−2
1
1
1
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
2
8
4
⎦
⎥
⎥
⎤
Consider, AX=B
On multiplying A
−1
both sides, we get
A
−1
.A.X=A
−1
.B
X=A
−1
.B .............(4)
Now,
A
−1
=
∣A∣
1
Adj(A)
∣A∣=
∣
∣
∣
∣
∣
∣
∣
∣
1
1
3
1
−2
1
1
1
1
∣
∣
∣
∣
∣
∣
∣
∣
∣A∣=6
Now, for Adj(A), we need minors and co-factors.
M
11
=−3,M
12
=−2,M
13
=7
M
21
=0,M
22
=−2,M
23
=−2
M
31
=3,M
32
=0,M
33
=−3
Therefore,
A
−1
=
6
1
⎣
⎢
⎢
⎡
−3
2
7
0
−2
2
3
0
−3
⎦
⎥
⎥
⎤
From equation (4),
X=
6
1
⎣
⎢
⎢
⎡
−3
2
7
0
−2
2
3
0
−3
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
2
8
4
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
−2
3
⎦
⎥
⎥
⎤
Hence, x=1,y=−2,z=3.